Problem: Factor the following expression: $5$ $x^2+$ $27$ $x+$ $10$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(5)}{(10)} &=& 50 \\ {a} + {b} &=& & & {27} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $50$ and add them together. The factors that add up to ${27}$ will be your ${a}$ and ${b}$ When ${a}$ is ${2}$ and ${b}$ is ${25}$ $ \begin{eqnarray} {ab} &=& ({2})({25}) &=& 50 \\ {a} + {b} &=& {2} + {25} &=& 27 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {5}x^2 +{2}x +{25}x +{10} $ Group the terms so that there is a common factor in each group: $ ({5}x^2 +{2}x) + ({25}x +{10}) $ Factor out the common factors: $ x(5x + 2) + 5(5x + 2) $ Notice how $(5x + 2)$ has become a common factor. Factor this out to find the answer. $(5x + 2)(x + 5)$